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    • CommentRowNumber1.
    • CommentAuthortomr
    • CommentTimeJun 16th 2018
    Let C be unspecified category and let we know that C have some universal properties, some limits or colimits, let we know that C can admit some algebraic structure (e.g. have monadic endofunctor) or C can be Eilenberg-Moore or Kleisli category (and so so, so on, I am only beginner at category theory and I guess that everyone else can name far more properties that categories can have).

    My question is - what can we deduce about objects and morphisms of this category from the properties of this catgory? Even more - maybe we can recover from the properties of the C the full structure of the objects and morphisms.

    E.g. each logic can be assigned the tuple called institution (e.g. https://kwarc.info/people/frabe/Research/GMPRS_catlog_07.pdf and https://www.amazon.co.uk/Institution-independent-Model-Theory-Studies-Universal-ebook/dp/B00KTHC9AI) which is tuple of categories and functors. There are catalogized around 2.000 different logics, not all of them have their institutions invented but maybe all they can have institutions. There is urgent need to devise new logics and to quickly check the properties of newly minted logics, i.e. are they complete and sound. E.g. there are dynamic action logics from the one side and there are linear logics with temporal, spatial, epistemic and some other modalities, but there is no good example of the logic that combines both aspects - dynamic actions and substructural modalities.

    So - maybe we can work at the institution level and devise in the first step the tuple of categories and functors with some properties that can go into the institution of the future logic and after then we can infere the exact objects and morphisms for the new minted categories. E.g. instutions have category of signature, so - maybe we can recover signature from the properties of the category.

    That is one example. But maybe there are more example where such approach can be desirable.

    I have asked the same question in math.stackexchange https://math.stackexchange.com/questions/2821628/algorithms-and-procedures-to-recover-objects-and-morphisms-from-the-properties-o but there is quite aggresive stance against it and it is being voted to be closed.

    I have to hear some thoughts about this deduction process. Maybe there are some good literature on this?